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Excitons in van der Waals materials: From monolayer to

bulk hexagonal boron nitride

Jaakko Koskelo, Giorgia Fugallo, Mikko Hakala, Matteo Gatti, Francesco

Sottile, Pierluigi Cudazzo

To cite this version:

Jaakko Koskelo, Giorgia Fugallo, Mikko Hakala, Matteo Gatti, Francesco Sottile, et al.. Excitons

in van der Waals materials: From monolayer to bulk hexagonal boron nitride. Physical Review

B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2017, 95 (3),

�10.1103/PhysRevB.95.035125�. �hal-02073979�

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Excitons in van der Waals materials: From monolayer to bulk hexagonal boron nitride

Jaakko Koskelo,1Giorgia Fugallo,2,3Mikko Hakala,1Matteo Gatti,2,3,4Francesco Sottile,2,3and Pierluigi Cudazzo2,3

1Department of Physics, P.O. Box 64, University of Helsinki, FI-00014 Helsinki, Finland

2Laboratoire des Solides Irradi´es, ´Ecole Polytechnique, CNRS, CEA, Universit´e Paris-Saclay, F-91128 Palaiseau, France

3European Theoretical Spectroscopy Facility (ETSF)

4Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-Yvette, France

(Received 29 September 2016; published 17 January 2017)

We present a general picture of the exciton properties of layered materials in terms of the excitations of their single-layer building blocks. To this end, we derive a model excitonic Hamiltonian by drawing an analogy with molecular crystals, which are other prototypical van der Waals materials. We employ this simplified model to analyze in detail the excitation spectrum of hexagonal boron nitride (hBN) that we have obtained from the ab initio solution of the many-body Bethe-Salpeter equation as a function of momentum. In this way, we identify the character of the lowest-energy excitons in hBN, discuss the effects of the interlayer hopping and the electron-hole exchange interaction on the exciton dispersion, and illustrate the relation between exciton and plasmon excitations in layered materials.

DOI:10.1103/PhysRevB.95.035125

I. INTRODUCTION

In many nanostructured materials, while strong covalent bonding provides the stability of the subnanometric elementary units, the whole assembly is held together by weak van der Waals interactions. The individual building blocks hence maintain most of their intrinsic characteristics also when arranged together to form a crystalline solid. In principle, novel materials properties can be thus tailored by controlling those of the elementary units. This bottom-up strategy in the synthesis of new materials has been intensively followed since the 1980s, when small atomic aggregates, nanoclusters, fullerenes, nanotubes, etc. started to attract enormous attention [1–4]. After the isolation of graphene in the mid-2000s, the focus of interest in nanotechnology applications largely shifted towards two-dimensional (2D) materials [5]. In recent years, monolayers or few-layer crystals of hexagonal boron nitride (hBN), black phosphorus, transition-metal dichalcogenides, and several other materials have also been heavily investi-gated [6,7]. The technological challenge now resides in the ability to stack together different atomically thin layers in order to build new kinds of “van der Waals heterostruc-tures,” with the goal of realizing devices with customized functionalities [8,9].

In order to design materials with desired features for improved nanoelectronics and optoelectronics applica-tions [10,11], the optical properties of layered materials need to be understood in detail. Due to the reduced effective screening [12], the optical response of 2D materials is domi-nated by strong electron-hole (e-h) interactions giving rise to bound e-h pairs, i.e., excitons. Nowadays, the state-of-the-art method to describe excitonic effects in condensed matter is the solution of the Bethe-Salpeter equation (BSE) [13,14] within the GW approximation (GWA) [15] of many-body perturbation theory [16]. As a matter of fact, in the last couple of decades, the ab initio BSE scheme [17–20] has been successfully applied to a wide variety of materials, including systems with reduced dimensionality [16,21,22].

Here, on the basis of ab initio GW-BSE calculations, we derive a general formalism to describe excitons in layered crystals, starting from the knowledge of the excitations of

a single layer. To this end, we proceed by analogy with molecular crystals [23–25], which can indeed be considered as the prototypical case of van der Waals materials. In this way, we obtain a general picture of excitonic effects in layered systems in terms of the interplay between e-h exchange interaction and band dispersion (i.e., interlayer hopping), which allows us to distinguish in a simple manner excitons of different character (e.g., intralayer and interlayer excitons). To numer-ically illustrate our analysis, we have chosen a prototypical layered material, namely hexagonal boron nitride, for which GW-BSE calculations are well established [26–36]. In hBN, the calculated dielectric function has already shown to be in excellent agreement with experiment in a wide range of energy and momentum [34]. Here we obtain the eigenvalue spectrum of the excitonic Hamiltonian as a function of momentum [37] and discuss its relation with quantities that are accessible via experiments.

The present work also extends to the exciton case (via the BSE formalism) the previous ab initio investigations that studied plasmons (i.e., collective electronic excitations) in prototypical layered systems such as graphite [38–40] or multilayer graphene [40,41]. In those materials, dielectric properties as a function of momentum q and interlayer distance d were calculated in the random-phase approxima-tion (RPA) within the framework of time-dependent density functional theory. Those studies already addressed general questions such as the effects of crystal local fields due to spatial inhomogeneities in the charge density variation of the Hartree potential, the role of the interlayer coupling due to the long-range Coulomb interaction between charge oscillations on different layers, and the possibility to adopt a local-response approximation to formally relate 2D and 3D response functions [40]. More recently, a “quantum-electrostatic heterostructure” model [42] was similarly derived to describe the dielectric properties of complex multilayers starting from those of the single-layer building blocks, also taking into account the long-range coupling between layers due to the Coulomb interaction. However, in both cases, hybridization effects were neglected: in the present work, they will be analyzed in detail in terms of interlayer hopping mechanisms.

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II. THEORETICAL FRAMEWORK AND COMPUTATIONAL DETAILS

The BSE is a formally exact Dyson-like equation relating the electron-hole correlation function L to its independent-particle version L0 [43]. Within the GWA to the self-energy, the BSE reads

L(1,2,3,4)= L0(1,2,3,4)+ L0(1,2,5,6) × [v(5,7)δ(5,6)δ(7,8)

− W(5,6)δ(5,7)δ(6,8)]L(7,8,3,4), (1) where we have used the shorthand notation (1) for position, time, and spin (r1,t11) and repeated indices are integrated over. In Eq. (1), v is the bare Coulomb interaction and W its statically screened version calculated at the RPA level. The former enters the kernel of the BSE (1) as an e-h exchange repulsive interaction and is responsible for crystal local-field effects. The latter is a direct attractive e-h interaction that is at the origin of excitonic effects, including the formation of bound excitons. For triplet excitons, the e-h exchange interaction v is absent.

The diagonal of the correlation function L yields the density-density response function χ (1,2)= L(1,1,2,2). In a crystal, by taking the Fourier transform of χ to frequency and reciprocal-lattice space, one directly obtains the loss function −Im−1 M as −Im−1 M (q,ω)= − q2Imχ (q,q,ω). (2) Here, M is the macroscopic dielectric function and q is a

wave vector such that q= qr+ G0, where qr belongs to

the first Brillouin zone (1BZ) and G0 is a reciprocal-lattice vector. The loss function, which can be measured by inelastic x-ray scattering (IXS) and electron energy loss spectroscopy (EELS), describes the longitudinal linear response of the sys-tem to an external potential. Hence it gives access to collective excitations such as plasmons, and (screened) electron-hole excitations.

Optical absorption spectra are related to the

vanishing-q limit of ImM(q,ω), which can be obtained from the Fourier transform of the modified response function ¯χ(1,2)=

¯

L(1,1,2,2),

ImM(q,ω)= −

q2Im ¯χ(q,q,ω), (3) where ¯L satisfies the BSE (1) with the modified Coulomb interaction ¯v at the place of v. In the reciprocal space, ¯v is defined to be equal to v, except for the G0component for which it is set to 0 [37]. Therefore, the difference between optical absorption and loss function at q= 0 is given by the long-range

G0= 0 component of the Coulomb interaction [22,44,45] [which is absent for ImMin the BSE (1)].

The loss function can also be explicitly written in terms of the imaginary and real parts of the macroscopic dielectric function, −Im−1 M (q,ω)= ImM(q,ω) [ReM(q,ω)]2+ [ImM(q,ω)]2 . (4)

Plasmon excitations are peaks in−ImM−1corresponding to the frequencies where ReMis zero and ImM(which provides the damping of the plasmon) is not too large.

In order to describe correlated e-h pairs explicitly, the BSE (1) (with ¯v at the place of v) can be cast in the form of a two-particle Schr¨odinger equation with an excitonic Hamiltonian, ˆ Hex(q)=  ck Eck+qack+qack+q−  vk Evkb†vkbvk +  vck, vck  2δmv¯vckvck(q)−W vck vck(q)  ack+qb†vkbvkack+q. (5) Here k, belonging to the 1BZ, and v(c) denote a valence (conduction) Bloch state of energy Evk(Eck) calculated within

the GWA; a†(a) and b†(b) are creation (annihilation) operators for electrons and holes, respectively; δmis 1 for the singlet and

0 for the triplet channel.

The first line of (5) is an independent particle Hamiltonian ˆ

Hip[corresponding to L0in the Dyson equation (1)], while the second line contains the interaction terms stemming from the kernel of (1). The matrix elements of ¯v and W are calculated in the basis of Bloch states as [21,37]

¯ vvckvck(q)=  drdrψck+q(r)ψvk(r) ¯v(r,r) × ψck+q(r)ψv∗k(r), (6) Wvvckck(q)=  drdrψck+q(r)ψck+q(r)W (r,r) × ψvk(r)ψv∗k(r). (7) In Eq. (5), we have adopted the Tamm-Dancoff approxima-tion (TDA), which amounts to neglecting antiresonant c→ v transitions and their coupling with resonant v→ c transitions (extension to the general case can be seen in [37]).

The macroscopic dielectric function,

M(q,ω)= 1 − q2  λ  vckvck(q) ˜ρvck(q)  2 ω− Eλ(q)+ iη , (8)

with the oscillator strengths ˜ρvck(q) defined as

˜

ρvck(q)= vk − qr|e−iq·r|ck, (9)

and the exciton wave function | λ(q) =

vck

Avckλ (q)ack†b†vk+q|0, (10)

where q is the total momentum of the two-particle state, can be thus written in terms of the eigenvectors Aλ(q) and the eigenvalues Eλ(q) of the excitonic Hamiltonian ˆH

ex (5),

ˆ

Hex(q)Aλ(q)= Eλ(q)Aλ(q). (11)

The excitonic eigenvalues Eλ(q) of ˆH

ex are hence the poles

of the ¯Land Mfunctions in the frequency domain. They give rise to peaks in the spectrum of ImM(q,ω) whose intensity is

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given by the numerator of Eq. (8). If it is negligibly small, the corresponding excitonic state is said to be dark.

The inverse macroscopic dielectric function M−1 can be analogously obtained from the eigenvectors and eigenvalues of the excitonic Hamiltonian Hex that, in addition to Hex (5),

also includes the long-range component of the Coulomb interaction, M−1(q,ω)= 1 + q2  λ  vckAλ vck(q) ˜ρvck(q)  2 ω− Eλ(q)+ iη . (12)

Therefore, the loss function−ImM−1(q,ω) can also be decom-posed in terms of the eigenvalues Eλ(q) and the eigenvectors

Aλ(q) of H ex.

In our first-principles calculations, we obtain the single-particle states ψnkusing Kohn-Sham (KS) density functional

theory within the local-density approximation (LDA) [46]. We use Troullier-Martins pseudopotentials [47], and expand the KS wave functions in a plane-wave basis set with a cutoff of 30 Hartree. The lattice parameters for the bulk are optimized using the LDA. We also consider hBN systems with variable interlayer distances d for which the in-plane lattice vectors are kept constant to the bulk value. On the basis of the results of GW calculations for hBN bulk [36], we apply a scissor operator of 1.96 eV to correct for the LDA underestimation of the single-particle band gap. For larger interlayer distances

d, the GW correction increases [29]: for example, it becomes 2.47 eV for d = 1.5d0, where d0is the interlayer separation of the bulk. For the GW-BSE computational details of the hBN monolayer, we refer to Ref. [35]. In all the other cases, we sample the Brillouin zone using a 48× 48 × 4 -centered grid. For the BSE calculations at finite q, we follow the same procedure as described in Ref. [34]. To simplify the analysis of the results in Sec.IV, here we use a minimal e-h transition basis set comprising two valence and two conduction bands and solve the BSE within TDA. As a consequence of the Kramers-Kronig relations, ReM converges more slowly than ImM with the number of higher-energy e-h transitions and (especially at small q) is affected by the coupling with antiresonant transitions [48] neglected in the TDA. While in the present case the main interest is to establish a direct connection between the electronic excitations characterizing ImM(q,ω) and the loss function−ImM−1(q,ω), for the comparison of the calculated loss-function spectra with experiment, we refer to Ref. [34]. In the construction of the BSE Hamiltonian, we expand the single-particle states and static dielectric function with plane-wave cutoffs up to 387 and 133 eV, respectively. We perform the KS and static screening calculations using

ABINIT[49], and BSE calculations with EXC[50]. All of the

spectra presented in the following sections are calculated for in-plane momentum transfer q along the M direction.

III. RESULTS

The two panels of Fig.1display the real and imaginary parts of the macroscopic dielectric function Mand the loss function −Im−1

M of the bulk crystal of hBN calculated by solving the BSE for two different in-plane momenta q. At vanishing q (top panel of Fig.1), the prominent peak at 5.67 eV in the absorption

FIG. 1. The real and imaginary parts of the macroscopic dielectric function ReMand ImMand the loss function−ImM−1calculated at

two different wave vectors q along the in-plane M direction. For improved visibility, the loss functions have been rescaled. The vertical arrows mark the smallest independent-particle GW transition energy (which for q= 0 corresponds to the direct band gap).

spectrum ImMis a tightly bound exciton, located well within the direct band gap [51] (which in GW amounts to 6.47 eV and is marked by the vertical arrow in the top panel of Fig.1). In the plot, we have labeled the main peak as A+(the explanation of the identification of the various excitations will be the subject of the detailed analysis in Sec.IV). Other structures are visible in the spectrum at higher energies, but for simplicity here and in the following we will focus on the lowest-energy excitations. As explained in previous works [28–31], the main absorption peak derives from π− π∗transitions between top-valence and bottom-conduction bands that are visible for in-plane light polarization [52]. Through the Kramers-Kronig relation, this A+peak of ImMinduces a strong oscillation in ReM, which crosses the zero axis with a positive slope at 5.99 eV. ImM being small at this energy, this zero of ReM gives rise to a plasmon resonance in the loss function−ImM−1, which shows a peak at the same energy [see Eq. (4)]. Here it is worth noticing that in hBN, this plasmon excitation also lies within the direct gap, since the collective charge excitation of the π electrons is strongly affected by the e-h attraction [34]. As discussed in details in Refs. [33,34], for increasing q, this π plasmon disperses to higher energies and, at larger q, it enters the continuum of particle-hole excitations.

As a matter of example, the bottom panel of Fig.1shows the spectra obtained for the second smallest finite q that we have considered in our calculations (for the other momentum

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FIG. 2. Exciton eigenvalue spectrum Eλ(q) in bulk hBN for in-plane q along M for (a) the electron-hole correlation function ¯L

corresponding to ImM(q,ω) featuring singlet excitons, (b) the electron-hole correlation function L corresponding to the loss function

−Im−1

M(q,ω), displaying plasmons and e-h excitations, and (c) triplet excitons. The M length is 1.45 ˚A −1

. For the explanation of the peak labels, see the main text. In the loss-function plot (b), the solid black line is a guide for the eye in order to better track the plasmon dispersion (corresponding to the A+feature).

transfers, not shown here, similar considerations can be made). Globally, the spectra at finite q remain qualitatively similar to the q= 0 case shown in the top panel of Fig.1. Still we can recognize that in ImM, a new small structure A−appears on the low-energy side of the main A+peak. The A−feature also induces a new shoulder in ReM in the same energy range. Finally, a new small peak “X” is visible in the loss function −Im−1

M at 5.88 eV, i.e., before the π plasmon. This new peak (which does not take place in correspondence with a zero of ReM, and hence is not a plasmon) matches a new very weak peak in ImM, so it has to be ascribed to a new many-body electron-hole excitation that becomes active at q= 0.

As discussed in Sec.II, the spectra for ImMand−ImM−1 can also be analyzed in detail by making use of the eigenvalues and eigenvectors of the excitonic Hamiltonian that enter Eqs. (8) and (12), respectively. Figure2(a)shows the 18 lowest energies Eλas a function of q for the singlet excitons that are

obtained from the diagonalization of excitonic Hamiltonian

Hex (5). They are hence the poles of M (8) and of the

modified two-particle correlation function ¯L. The color scale represents their intensity|Aλ· ˜ρ| at the numerator of Eq. (8).

Red squares are for states that have a visible peak in ImM, while blue squares are dark exciton states with no intensity in the spectrum. The other two panels of Fig.2use the same

repre-sentation. Figure2(b)displays the exciton eigenvalues Eλ(q)

obtained from the diagonalization of Hex that includes the

long-range Coulomb interaction: they enter the loss-function spectra −ImM−1(q,ω) (12). Finally, in Fig. 2(c), the triplet exciton energies are also reported for comparison (they cannot be directly measured by loss or absorption spectroscopies). They are calculated from the excitonic Hamiltonian Hex (5),

where the e-h exchange interaction ¯v is absent. With respect to the singlet excitons, the triplet energies are globally lower [compare Figs.2(a)and2(c)], as the e-h exchange interaction is repulsive and hence yields singlet states that have higher energies than the corresponding triplets.

The first and third q points in Figs.2(a)and2(b)allow us to better understand the absorption and loss spectra plotted in the two panels of Fig.1. For example, in Fig.2(a), we discover that in the optical limit q→ 0, the first visible exciton A+ is degenerate with a dark state [53] (labeled B+ here) and that below them there are two other degenerate dark excitons A− and B− that do not contribute to the q→ 0 absorption spectrum in the top panel of Fig.1 (this point was already the subject of discussion in Refs. [28,30,31]). We can also see that at finite q, one of the two lowest dark excitons becomes visible, giving rise to the low-energy peak A−in the absorption spectrum of Fig.1, bottom panel. Finally, the weak peak X at

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5.88 eV is due to another exciton state that is dark at q= 0 and switches on at q= 0. For all wave vectors q, at higher energies the exciton states become very dense, forming a continuum of excitations. In Sec.IV, we will focus on the four lowest-energy discrete states that are well within the fundamental gap.

We can now repeat the same analysis for the loss functions −Im−1

M (q,ω) in Fig.1 using the poles of L represented in Fig.2(b). We thus discover that at q= 0, the plasmon excita-tion at 5.99 eV is not the lowest-energy eigenvalue. It is actually already located in the energy region where e-h excitations are rather dense. So it is not easy to track its dispersion after the first few q points. At the bottom of the eigenvalue spectrum, there are instead three dark states (two of them are degenerate at q= 0) that are well separated from the other excitations. They have a dispersion as a function of q that is similar to that of the lowest poles of ¯Lin Fig.2(a). It is hence tempting to make a connection between them. In Sec.IV, we will explain rigorously why this is indeed the case (so they are labeled A− and B± here) and why the plasmon excitation instead has a A+ character. Finally, at q= 0 at 5.88 eV, we recognize the same X excitation that is also present in the spectrum of ¯L

in Fig.2(a)and is responsible for the weak structures in the absorption and loss spectra in the bottom panel of Fig.1.

IV. DISCUSSION

A. The exciton Hamiltonian in layered crystals In order to interpret the numerical results of the previous section, here we generalize the approach that some of us

introduced in Ref. [23] to explain the excitonic properties of molecular crystals. We thus rewrite the excitonic Hamiltonian

ˆ

Hex [which in Eq. (5) is expressed in terms of Bloch wave

functions delocalized all over the crystal] in the basis of wave functions localized on the elementary units of the system. While in molecular crystals the elementary units are the single molecules, in the present case they are the single layers of BN (stacked along the z axis). We assume that the one-particle wave functions ψ(r) localized on different layers do not overlap and can be factorized in in-plane φ(ρ) and out-of-plane χ (z) components, with r= (ρ,z). Specifically, for given in-plane wave vector k and out-of-plane kz, the single-particle

wave function ψk,kz(r) is expanded in the basis of single-layer

wave functions as RicikeikzRφki(i)(ρ)χi(z− R). Here R is the lattice vector along z and the index i denotes the layers inside the unit cell. We also consider the possibility that the various layers stacked along z are rotated one with respect to another by an angle β (in hBN, β= 60◦), and therefore also the 2D first Brillouin zones are rotated by an angle β [54]. Hence, choosing a reference layer i= 1, we define k(i)= k for i= 1 and k(i)= β−1k for i= 1, with βk being the wave vector obtained rotating k by an angle β (see AppendixA

for more details). For simplicity, we further consider for each layer a two-band system, with only one valence v and one conduction c band. Under these assumptions, the whole excitonic Hamiltonian ˆHex of the crystal given by

Eq. (5) takes the simple form of the sum of three terms ˆ Hex = ˆHip+ ˆKCT + ˆKF R: ˆ Hip=  k1k2  RS  ij EcRiSj(k1,k2)ack1Riack2Sj−  k1k2  RS  ij ERiSjv (k1,k2)bvk1Ribvk2Sj, ˆ KF R=  k1k2k3k4  Ri,Sj  ¯

vSj,SjRi,Ri(vk1ck2vk3ck4)− δRi,SjWRi,RiRi,Ri(vk1ck2vk3ck4)  ack 2Rib vk1Ribvk3Sjack4Sj, ˆ KCT = −  k1k2k3k4  Ri,Sj (1− δRi,Sj)W Sj,Ri Sj,Ri(vk1ck2vk3ck4)ack2Rib vk1Sjbvk3Sjack4Ri, (13) with WSj,RiSj,Ri(vk1ck2vk3ck4)=  drdrφcki∗ 2(ρ)χ ic (z− R)χ jv (z− S)φ j vk3(ρ )W (r,ri ck4(ρ)χ i c(z− R)φ j vk1(ρ j v(z− S), (14) ¯ vSj,SjRi,Ri(vk1ck2vk3ck4)=  drdrφick∗ 2(ρ)χ ic (z− R)φ j vk3(ρ jv (z− S)¯v(r,r j ck4(ρ j c(z− S)φ i vk1(ρ)χ i v(z− R). (15)

In the Bloch picture, ˆHip contains independent e-h

tran-sitions between single-particle bands. Equivalently, here ˆHip

describes scattering processes from layer to layer, indepen-dently for electrons and holes, being

EvRiSj(c) (k1,k2)= Evi(c)(k1Ri,Sjδk1,k2+ t

v(c)k1,k2

Ri,Sj , (16)

where Eiv(c)(k1) is the single-layer band dispersion and t

v(c)k1,k2

Ri,Sj

are interlayer hopping matrix elements (see AppendixA) that give rise to the finite kzdispersion of the bands in the crystal

(see Fig.8in AppendixB).

In Eq. (13), the second and third terms ˆKF R and ˆKCT

describe the interaction between an electron and a hole that are

localized on the same layer or on different layers, respectively. In order to keep a closer contact with the exciton physics of molecular crystals, here we name the intralayer configuration as a “Frenkel” (FR) exciton and the interlayer configuration as a “charge-transfer” (CT) exciton. In other words, in the present context, we call FR an exciton that is fully localized on a single layer, independently of being localized or not within the layer. Therefore this definition applies equivalently for excitons with different in-plane localization characters, as, for example, in hBN (where the exciton is tightly bound also within the single layer [28]) or in MoS2 (where it is weakly bound [55]). We note that the e-h exchange interaction ¯

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the same layer, and therefore it is absent for CT excitons in Eq. (13).

The FR and CT interaction terms in Eq. (13) are coupled by the interlayer hopping terms in ˆHip. Without the interlayer

hopping tv(c)k1,k2

Ri,Sj , the excitonic Hamiltonian (13) factorizes

into two independent blocks: a CT Hamiltonian ˆHCT = ˆHip +

ˆ

KCT describing an interacting e-h pair localized on different

layers and a FR Hamiltonian ˆHF R = ˆHip + ˆKF R describing

an interacting e-h pair on the same layer (in both cases, we set

tv(c)k1,k2

Ri,Sj = 0 in Hip).

The CT exciton wave functions [56],  exCT(q)= λ  ij τ cijτ ij,λτ(q), (17) with  ij,λτ(q)=√1 N  R  k

Aλ,ij,vc,kτ(q)ack†(i)Rib†vk(j )+q(j )R+τj|0, (18) are already the eigenfunctions of ˆHCT that can be directly

built from the excitations of the single layers. The Frenkel Hamiltonian ˆHF Rinstead also contains an interlayer coupling

that needs additional consideration.

By further splitting the e-h exchange interaction ¯v into a long-range contribution ¯v0 (corresponding to the G||= 0 component in reciprocal space) and a short-range contribution ¯¯v such that ¯v= ¯v0+ ¯¯v, the FR Hamiltonian ˆHF R can be

separated into an intralayer term ˆHLand an interlayer coupling

ˆ¯ V, ˆHF R = ˆHL+ ˆ¯V , with ˆ HL=Hip +  k1k2k3k4  Ri  ¯¯vRi,Ri Ri,Ri(vk1ck2vk3ck4)− W Ri,Ri Ri,Ri(vk1ck2vk3ck4)  ack 2Rib vk1Ribvk3Riack4Ri, (19) ˆ¯ V =  k1k2k3k4  Ri,Sj ¯ vSj,Sj0Ri,Ri(vk1ck2vk3ck4)ack2Rib vk1Ribvk3Sjack4Sj, (20) where ¯¯vSj Sj RiRi(vk1ck2vk3ck4)=  q|| δk 1,k2+q(i)||δk3,k4+q(j )||  qzGz  G||=0 |q + G|2ρ˜ i ck2vk1(q (i) || + G(i)|| ) ˜ρckj∗4vk1(q (j ) || + G(j )|| ) ×  dzχc(z)ei(qz+Gz)zχ v(z)  2e−iqz·(S−R)e−i(qz+Gz)dij, (21) ¯ v0RiRiSj Sj (vk1ck2vk3ck4)=  q|| δk 1,k2+q(i)|| δk3,k4+q(j )||  qz,Gz= 0, G||= 0 |q + G|2ρ˜ i ck2vk1(q (i) || ) ˜ρckj∗4vk1(q (j ) || ) ×   dzχc(z)ei(qz+Gz)zχ v(z)  2e−iqz·(S−R)e−i(qz+Gz)dij, (22) ˜ ρckvki (q(i)|| + G(i)|| )= 

dρφick∗(ρ)ei(q(i)||+G(i)||)·ρφi

vk(ρ). (23)

Equivalently, Eq. (21) can be written in terms of the partial Fourier transform of the Coulomb potential [40],

v(q||+ G||,z,z)= |q||+ G|||e−|q||+G||||z−z | , (24) as ¯¯vSj Sj RiRi(vk1ck2vk3ck4)=  q|| δk 1,k2+q(i)|| δk3,k4+q(j )||  G||=0 |q||+ G|||˜ ρcki 2vk1(q (i) || + G(i)||) ˜ρckj∗4vk1(q (j ) || + G(j )|| ) ×  dz  dzχc(z)χv(z)e−|q||+G||||z−z | χv(z)χc(z)e−|q||+G||||S−R|e−|q||+G|||dij. (25)

From Eq. (25), we can conclude that the off-diagonal elements

S= R and i = j of ¯¯vSj SjRiRi are actually zero, for the presence of the exponential terms e−|q||+G||||S−R|with G||= 0. For its short-range nature, the ¯¯vinteraction therefore does not couple different layers.

With respect to the Bloch picture, such a transformation and decomposition of the excitonic Hamiltonian (5) illustrates much more clearly the physics of excitons in layered materials that we want to uncover. Here the eigenstates of ˆHLrepresent

the excitations of an elementary unit of our van der Waals material, namely a single BN layer embedded in the bulk

crystal. They are formally analogous to the excitations of a single molecule in a molecular solid. Thus, by analogy with molecular crystals, a FR exciton in the present case can be seen as an elementary excitation of a single layer, which can scatter from one layer to another due to the interlayer coupling ˆ¯V. From a mathematical point of view, this means that we expand the FR exciton wave functions (which are the eigenfunctions of ˆHF R) on the basis of the eigenstates of ˆHL:

 exF R(q)=√1

N



λ,Ri

(8)

where

 R,iiλ (q)=

k

Aλ,ivc,k(q)ack†(i)Rib†vk(i)+q(i)Ri|0, (27)

and where we have used the fact that for in-plane q one has

eiq·R= 1. The matrix elements of ˆ¯V are

R,iiλ (q) ˆ¯V Rλ,jj(q)  =  Gz=0,G||=0 |q + G|2  Siλ(q)∗Sjλ(q) ×  dzχc(z)eiGzzχ v(z)  2e−iGzdij, (28) where Siλ(q) is the oscillator strength of the exciton λ of the

layer i,

Siλ(q)=

k

Aλ,ivc,k(q(i)) ˜ρvcki (q(i)), (29)

and where we have used the fact that χni(z)= χ j

n(z− dij)≡

χn(z− dij) for both n= v,c, with dij the distance between

the layers i and j . From Eq. (28), we realize that ˆ¯V operates only on visible excitons and cannot couple visible and dark excitons for which Sλ

i(q)= 0.

B. The exciton Hamiltonian in hBN

If we consider a crystal with two inequivalent layers per unit cell, as is the case for hBN, for each quantum number λ that defines an excitation of the single layer one has four excitons in the bulk [we take into account only first nearest-neighbor CT excitons and assumeτ = 0 in Eq. (17)]. The FR and CT excitons that diagonalize the excitonic Hamiltonian ˆHex (13)

in the absence of interlayer hopping are then the symmetric and antisymmetric combinations with respect to the exchange of the e-h pair between two inequivalent layers,

|CTλ ± = 1 √ N  R 1 √ 2 λ R,12  ± R,21λ , (30) |F Rλ ± = 1 √ N  R 1 √ 2  R,11λ ± R,22λ . (31) The|CT±λ states are degenerate, while the energy separation between the|F Rλ

± states in the context of molecular crystals is usually called Davydov splitting [57].

In the case of hBN, the two lowest excitons of the BN monolayer, which are degenerate at q= 0, are a visible exciton

Aand a dark exciton B [see Fig.4(a)]. They originate from transitions from the top-valence to the bottom-conduction bands with k vectors located around the K or Kpoints of the Brillouin zone, respectively [35,58]. These two single-layer

A and B excitons hence produce eight excitons (four FR and

four CT excitons) in the bulk crystal. Since the B exciton is dark for all q along M, in the bulk the A and B excitons are not mixed by ˆ¯V [see Eq. (28)] and preserve their identity. The four|CT±A,B excitons are located at higher energies since they have smaller binding energies, as a result of the e-h attraction being smaller for interlayer e-h pairs than for intralayer e-h

pairs. In the following, we focus on the four|F R±A,B excitons that are the lowest-energy excitations in the bulk.

The A exciton of the single layer of energy EA(q)=

E11A(q)= EA22(q) gives rise to the two|F R±A excitons,

EA±(q)= EA(q)+ ¯IA(q)± ¯JA(q), (32) where ¯IA(q)± ¯JA(q) are the contribution to the exchange e-h

interaction ˆ¯V for the symmetric (+) and antisymmetric (−)

states, respectively. They are the excitation-transfer interac-tions that are responsible for the interlayer propagation of the FR exciton in the crystal [23]: ¯JAis related to the scattering

process of an e-h pair between two inequivalent layers and, analogously, ¯IA between equivalent layers in different unit cells. Explicitly, they read

¯ IA(q)=  Gz=0,G||=0 |q + G|2|S A(q)|2 ×  dzχc(z)eiGzzχ v(z)  2, (33) ¯ JA(q)=  Gz=0,G||=0 |q + G|2|S A(q)|2 ×   dzχc(z)eiGzzχ v(z)  2e−iGzd. (34)

We note that ¯IA(q) and ¯JA(q) are both zero at q= 0, since

the oscillator strength SA(q) in the dipole limit q→ 0 is

proportional to q· μA. Therefore, ¯I and ¯Jin layered systems do not yield any Davydov splitting between symmetric and antisymmetric excitons at q= 0, in contrast to the molecular

(a)

(b)

FIG. 4. Dispersion of (a) singlet and (b) triplet excitons in the BN single layer. The color key is the same as in Fig.2.

(9)

crystal case [23]. The matrix elements for r= (ρ,0) are F RA ±|eiq·r|0 = |SA(q)|2 ×  dzχc(z)χv(z)±  dzχc(z)χv(z) . (35) The symmetric |F RA

+ exciton is hence visible, while the antisymmetric|F RA

− exciton is dark, since the two integrals in Eq. (35) exactly cancel in this case. For the B exciton, the matrix element of ˆ¯V is zero [since SB(q)= 0 in Eq. (28)].

Therefore, the two|F RB

± excitons remain degenerate in the bulk,

E±B(q)= EB(q). (36) Moreover, as SB(q) is zero, they are both dark [see Eq. (35)].

In summary, by neglecting the interlayer hopping terms in the exciton Hamiltonian (13), we would expect that the two lowest A and B excitons of the BN single layer give rise to three FR dark excitons and one FR visible exciton in the bulk (together with CT excitons at high energies).

The effect of the hopping is, in general, to couple FR and CT excitons. This coupling produces states with mixed character, FR+CT and CT+FR, respectively, and modifies their energies. In hBN, as demonstrated in Appendix A, at q= 0 one has CTλ

±| ˆT |F Rλ = 0: excitons with different parities do not couple, giving rise to|(F R + CT )λ

± states with well-defined parity. Moreover, sinceCTλ

+| ˆT |F R+λ = CTλ| ˆT |F Rλ− (see AppendixA), at q= 0 the hopping induces a finite Davydov splitting between symmetric and antisymmetric excitons. Instead, at finite q, the various excitons formally lose their parity character as FR and CT states with different parities are generally allowed to mix together.

C. Exciton dispersion: Electron-hole exchange and interlayer hopping

On the basis of the previous analysis, we can now examine in detail the properties of the four lowest-energy singlet exci-tons in hBN [see Fig.2(a)]. In particular, we can understand the effect of e-h exchange by comparing singlet and triplet excitons [see Figs.2(a)and2(c)] because in the latter there is no e-h exchange. Moreover, we can also suppress the interlayer hopping by artificially increasing the interlayer distance d. The singlet and triplet exciton band structures obtained with

d = 1.5d0, where d0is the experimental interlayer distance of hBN, are displayed in Figs.3(a)and3(c). With this increased separation between BN layers, the interlayer hopping is reduced so much that the kzdispersion of the top-valence and

bottom-conduction single-particle bands becomes negligible around K (see AppendixB).

At q= 0, the four lowest singlet excitons are grouped in two pairs [see Fig.2(a)]. Since in the single layer the A and

B excitons are degenerate at q= 0 [see Fig. 4(a)] and the e-h exchange terms ¯I(q= 0) and ¯J(q = 0) are zero for all of them [see Eqs. (33) and (34)], the energy splitting between the two pairs must derive from the interlayer hopping (which we reasonably assume to be the same for A and B excitons). At

q= 0, the hopping conserves the parity character, removing

the degeneracy between symmetric and antisymmetric states. Indeed, this energy splitting is present also for the triplet exci-tons [see Fig.2(c)], whereas it becomes zero for an increased interlayer distance [see Fig.3(a)and3(c)]. Therefore, we can conclude that at q= 0, the two excitons of the lowest pair, which are both dark, are the antisymmetric |(F R + CT )A and|(F R + CT )B

− states, while the two excitons of the other pair are the symmetric|(F R + CT )A

+ (which is visible) and |(F R + CT )B

+ (which is dark). For simplicity, in Figs. 2 and3 we have labeled A± and B±, respectively, the states |(F R + CT )A

± and |(F R + CT )B±.

Having established the character of the excitons at q= 0, we can now track their dispersion as a function of q. The fact that one of the excitons of the lowest pair that is dark at q= 0 becomes visible at q= 0 for both the singlet and triplet cases [see Figs. 2(a) and 2(c)] is another effect of the interlayer hopping that at q= 0 mixes FR and CT states with different parities. This means that the parity is no longer a good quantum number and the eigenstates of the excitonic Hamiltonian are combinations of|(F R + CT ) and |(F R + CT )+ states. In this way, the dark exciton|(F R + CT )A

− is switched on by the effective coupling with the visible exciton|(F R + CT )A

+. Formally, all the excitons lose their defined parity, but here for simplicity we still call them A± (the two visible states) and B±(the two dark states).

In order to infer the effect of the interlayer hopping on the exciton dispersion, we compare the behavior of the triplet excitons in the bulk [see Fig.2(c)], for the increased interlayer distance [see Fig.3(c)] and in the monolayer [see Fig.4(b)]. In the monolayer, the A and B triplet excitons are almost degenerate: there is a tiny separation due to the direct e-h attraction W [35]. The same holds for d= 1.5d0, where there is no effect of the interlayer hopping. In the bulk, instead, the hopping acts differently for the various excitons, giving rise to a finite dispersion that removes the degeneracies. The energy-level ordering remains the same for all q. From the bottom to the top, one has the following states: B−, A−, B+, and A+.

The difference between the dispersions of the singlet and the triplet excitons in Figs.2(a)and2(c) illustrates the role of the e-h exchange interaction in the bulk as a function of

q. While the B± excitons keep the same dispersion in the two channels [as the e-h exchange ¯I(q)= ¯J(q) = 0 for B excitons], for the A±excitons we observe that the effect of the e-h exchange is larger for small q than for large q, where the dispersion of singlet and triplet excitons tends to be the same, being determined by the single-particle band dispersion only.

At increased interlayer distance d= 1.5d0, in contrast to the bulk, both for the singlet and the triplet channels also at finite q there remain one visible and three dark excitons, as in the limit q= 0 [see Figs. 3(a) and 3(c)]. This confirms that by suppressing the interlayer hopping, the antisymmetric|F RA

− exciton cannot couple with excitons of different parity and continues to be dark. All the excitons keep the same parity as at q= 0. The two dark |(F R + CT )B

± excitons remain degenerate, since the e-h interaction ˆV has no effect on them. They are located at lower energies than the |(F R + CT )A

± excitons as ˆV is repulsive. In particular, the dark|(F R + CT )A

(10)

FIG. 5. Dispersion of the visible A+exciton for different inter-layer distances d (d0is the experimental value). For each case, the

exciton energies are defined with respect to the corresponding q= 0 value.

|(F R + CT )A

+, implying that the effect of ¯I − ¯J is larger than ¯

I+ ¯J. In general, the energy-level ordering is, from the bottom

to the top,|(F R + CT )B

± (degenerate), |(F R + CT )A+, and |(F R + CT )A

−.

By increasing d, the screening of the e-h attraction W is reduced and, as a consequence, the binding energies of all the excitons increase (however, their absolute positions remain almost constant [29]). In order to directly compare, for increasing interlayer distances d, the dispersion of the visible A+exciton as a function of q= 2π/λ, in Fig.5we have hence aligned, for the different separations d, the exciton energies to their q = 0 value. By increasing the interlayer distance, the dispersion becomes steeper at small q and tends to be the same at large q. As a result of the competition between the e-h exchange interaction and the single-particle band dispersion, in the exciton dispersions we can always distinguish two regimes: (i) at large q (i.e., for λ d) the sum over Gzin Eq. (33) can

be approximated with an integral. So ¯IAand ¯JAbecome ¯ IA(q) q β(q)|S A(q)|2, (37) ¯ JA(q)≈ ¯IA(q)e−qd, (38) with β(q)=  dzχc(z)e−qzχv(z)  2. (39) Under these conditions, as shown in Ref. [35], ¯IA reaches

a constant value at large q. Moreover, since λ d, the exponential factor in Eq. (38) goes to zero and ¯JAbecomes negligible. As a consequence, in this regime the dispersion of the symmetric and antisymmetric excitons becomes the same and, at large q, is set by the hopping only. (ii) At small q (i.e., for λ d), the sums over Gzin Eqs. (33) and (34) is

independent of q and SA(q)= q · μA: ¯I and ¯J are quadratic in q. Therefore, in this regime, the exciton dispersion is also determined by the e-h exchange ˆ¯V, in addition to the hopping contribution that is always present. At small q, the e-h exchange interaction becomes more and more important as d increases, until in the 2D limit it becomes the dominant

FIG. 6. Energy difference between the plasmon and the visible A+ exciton as a function of momentum q for different interlayer distances.

contribution [compare the dispersion of the singlet in Fig.4(a)

and triplet in Fig4(b)]. Indeed, in the 2D limit, when Eqs. (37) and (38) are exact for every q, the e-h exchange contribution becomes linear in q, as explained in detail in Ref. [35].

D. Plasmon dispersion: Long-range Coulomb interaction In order to describe the plasmon properties, in the excitonic Hamiltonian (13) one has to replace the short-range ¯vwith the full Coulomb interaction v. This implies that in the long-range

G||= 0 contribution to the e-h exchange (22), the Gz= 0

component also has to be included. The excitation-transfer interactions [with ¯I and ¯J defined in Eqs. (33) and (34)] thus become I(q)=|S(q)|2 q2   dzχc(z)χv(z)  2+ ¯I(q), (40) J(q)=|S(q)|2 q2   dzχc(z)χv(z)  2+ ¯J(q). (41) The long-range contribution of the Coulomb interaction is responsible for the difference between the excitation spectra of ¯L and L, which are displayed in Figs. 2(a) and 2(b), respectively. By comparing the poles of ¯L and L, we note that the long-range term of v has no effect on the lowest excitons|(F R + CT )B

±, since I = J = 0 for them, and on the antisymmetric exciton|(F R + CT )A

−, as it exactly cancels in the difference I− J [see Eqs. (40) and (41)]. The repulsive long-range interaction is felt only by the symmetric state |(F R + CT )A

+ that is the plasmon excitation in L. As a consequence, its energy at q= 0 is upshifted with respect to the corresponding A+ pole of ¯L by ∼ (8π/q2)S(q= 0)= 8π|ˆq · μ|2. At finite q, the plasmon energy displays a quadratic dependence on q. Without interlayer hopping (i.e., for interlayer spacing d > 1.5d0), the plasmon dispersion is hence similar to that of the triplet exciton energy. This is a consequence of the cancellation at finite q occurring to a large extent between the first and second terms in Eqs. (40) and (41). While the first terms account for the difference between plasmon and singlet exciton (see Fig.6), the second terms are responsible for the difference between singlet and

(11)

FIG. 7. Energy difference between the singlet and triplet A+ excitons as a function of momentum q for different interlayer distances.

triplet excitons (see Fig.7). As a matter of fact, by comparing Figs. 6 and7, we notice that for each interlayer separation, they have an opposite behavior as a function of q.

At large q (i.e., for λ d), the long-range contribution becomes negligible. As shown in Fig. 6, for increasing d the plasmon energy approaches the visible-exciton energy for smaller and smaller q: the loss function −ImM−1 becomes equal to ImM when qd 1. In the 2D limit (i.e., d → ∞), as for any completely isolated system [44,45],−ImM−1 and ImMmathematically coincide for all q.

V. SUMMARY

From the solution of the ab initio Bethe-Salpeter equation (BSE) as a function of momentum q, we have obtained the eigenvalue spectrum of the excitonic Hamiltonian for the electronic excitations of hexagonal boron nitride and we have established the connection with measured optical absorption and energy-loss spectra. We have discussed the properties of both visible and dark excitons on the basis of a simplified model that we have derived from the full ab initio BSE and by analogy with the case of molecular solids. This model has allowed us to provide an efficient description of the excitations in the bulk crystal starting from the knowledge of the excitons in the single layer. In this way, we have obtained a general picture of the exciton physics in layered materials. Our analysis uncovers the interplay between the electronic band dispersion and the electron-hole exchange interaction in setting the exciton properties in this important class of materials. Holding a general validity, it can be similarly applied to other van der Waals systems.

ACKNOWLEDGMENTS

This research was supported by the MATRENA Doctoral Programme and Academy of Finland (Contract No. 1260204), by an ´Energies Durables Research Grant from the ´Ecole Polytechnique, the ´Ecole Polytechnique Foundation, and the EDF Foundation, by a Marie Curie FP7 Integration Grant within the 7th European Union Framework Programme, and by the European Union’s Horizon 2020 research and inno-vation programme under the Marie Sklodowska-Curie Grant

Agreement No. 660695. Computational time was granted by GENCI (Project No. 544) and the CSC - IT Center for Science.

APPENDIX A: CHARGE-TRANSFER AND FRENKEL EXCITONS IN hBN

In our model, we start from the assumption that the effective single-particle Hamiltonian that defines the electronic band structure of a layered system can be written as the sum of a single-layer Hamiltonian ˆHL(ρ,z) and an effective

out-of-plane potential δ ˆU(z), which describes the crystal field along the z axis. Under these conditions, the band index n and the in-plane wave vector k that define the eigenstates of ˆHL(ρ,z)

are also good quantum numbers for the bulk wave function

ψnk,kz (kz being the corresponding out-of-plane wave-vector

component in the 3D Brillouin zone). Hence, ψnk,kz can be

expanded in terms of φnki (ρ)χ i

n(z− R) (here, i denotes the

layer in the unit cell and R the lattice vector along z). For a system characterized by two layers per unit cell,

φi nk(ρ)χ

i

n(z− R) is a set of 2N degenerate states (N is the

number of unit cells) corresponding to the eigenvalues En(k)

of ˆHL(ρ,z) and represents a complete basis set for the bulk

wave function. Moreover, in the case of hBN with the AB stacking, the two layers in the unit cell are rotated one with respect to the other by an angle β= 60◦ and, therefore, the corresponding 2D first Brillouin zones are also rotated by the angle β. For a given wave vector k, the in-plane components of the electronic wave functions associated to two inequivalent layers are related by

φnk2 (ρ) = φn1βk(ρ), (A1)

whereβk is the wave vector obtained rotating k by an angle

β. Similarly for the corresponding eigenvalues, one has

E2n(k)= En1(βk). (A2)

Choosing the wave vector k in the first Brillouin zone of the reference layer i= 1, the single-layer basis set is split in two subsets of N wave functions: φ1nk(ρ)χn1(z− R) with energy

E1

n(k) and φn2β−1k(ρ)χn2(z− R) with energy En2(β−1k). The

ensemble of the two subsets represents a complete basis set for the representation of the bulk wave functions. In a more compact notation, the single-layer basis for both excitonic and single-particle Hamiltonians is given by the wave functions

φnki (i)(ρ)χ

i

n(z− R) with k(i)= k for i = 1 and k(i)= β−1k

for i= 2.

The single-particle Hamiltonian (written in second quanti-zation) hence takes the form

ˆ H = nkk  RS  ij

EnRiSj(k,k)ankRi ankSj, (A3)

where ERiSjn (k,k) are the matrix elements of HˆL(ρ,z) +

δ ˆU(z) and are given by the expression

EnRiSj(k,k)= Ein(k)δRi,Sjδk,k+ tnk,k



Ri,Sj, (A4)

with tRi,Sjnk,k denoting the effective interlayer hopping,

tRi,Sjnk,k=  dρφin,k∗ (ρ)φn,kj (ρ)  dzχni(z−R)δU(z)χnj(z− S). (A5)

(12)

Defining

tRi,Sjn =



dzχni(z− R)δU(z)χnj(z− S), (A6)

we have tRi,Sink,k = tn

Ri,Siδk,kfor i= j and t

nk,k

Ri,Sj = t

n

Ri,Sjδβ−1k,k

for i= j. We note that in the present case, the hopping tRi,Sjnk,k is not diagonal in k and, in this way, the single-particle energies

EnRiSj(k,k) in (A3) acquire a dependence on both k and k.

We consider a two-band system (n= c,v) and we take into account only the interlayer hopping between first-nearest-neighbor layers (i= j). In this case, the hopping operators acting on electrons and holes are given by the following expressions: ˆ Tc=  R,k tc[ackR 1acβ−1kR−τ2+ acβ−1kR2ackR1], (A7) ˆ Tv=  R,k tv[b†ckR1bcβ−1kR−τ2+ b†cβ−1kR2bckR1], (A8)

where τ is the smallest lattice vector (0,0,1) and tc(v)=

tR1,Rc(v) −τ2= tR2,R1c(v) . The effect of the hopping is to induce a dispersion along the z axis in reciprocal space and a splitting of the single-layer bands without modifying their in-plane dispersion. This is a consequence of the decoupling approximation between in-plane and out-of-plane coordinates. It is justified by the fact that in hBN, the excitons originate from a limited area in the Brillouin zone, so that we can assume that the kz dispersion in the single-particle band structure is

constant for all of the relevant k points.

First of all, we neglect the hopping terms in such a way that the charge-transfer and Frenkel excitons are decoupled [see Eq. (13)]. Here we analyze the interlayer charge-transfer exciton state, where the electron and the hole are localized on different layers. The charge-transfer wave function for the exciton state λ is  ij,λτ(q)= √1 N  R  R,Rλ,ij(q), (A9) with  R,Rλ,ij(q)= k

Aλ,ij,vc,kτ(q)ack†(i)Rib

vk(j )+q(j )R+τj|0, (A10) where k(i)= k for i = 1 and k(i)= β−1k for i= 2 (the same applies for the wave vector q), while the coefficients Aλ,ij,vc,kτ(q) satisfy the excitonic eigenvalue equation,

 Eic(k(i))− Evj(k(j )+ q(j ))Aλ,ij,vc,kτ(q)− k WRR+τjRi+τj,Ri(vk(j ) + q(j )ck(i)vk(j )+ q(j )ck(i) )Aλ,ij,vc,kτ(q) = Eλ ij,τ(q)A λ,ij,τ vc,k (q). (A11)

Here, i identifies the index of the layer where the electron of the CT e-h pair is located, while j the layer of the corresponding hole;τ defines the lattice-vector separation along z of the two unit cells to which the layers i and j belong. In the following, we will focus on the first-nearest-neighbor CT states for which

i= j and τ = (0,0,0) [for the other first-nearest-neighbor CT

state,τ would be (0,0, − 1)]. In this case, we have two possible

configurations for the e-h pair: i= 1 and j = 2 or i = 2 and

j = 1. They are described, respectively, by the equations

[Ec(k)− Ev(β−1k+ β−1q)]A λ,12 vc,k(q) − k W(vβ−1k+ β−1qckvβ−1k+ β−1qck)Aλ,vc,k12(q) = Eλ 12(q)A λ,12 vc,k(q), (A12) [Ec(β−1k)− Ev(k+ q)]Aλ,vc,k21(q) − k W(vk+ qcβ−1kvk+ qcβ−1k)Aλ,vc,k21(q) = Eλ 21(q)A λ,21 vc,k(q), (A13)

where we have dropped the indices i,j since the functional form of both the single-particle energies Ev and Ec and the

interlayer effective electron-hole interaction W is invariant under the exchange of the layer index. By applying the rotation

β to the k space, Eq. (A12) becomes [Ec(βk) − Ev(k+ q)]Aλ,vc,12βk(βq) − k W(vk+ qcβkvk+ qcβk)Aλ,vc,12βk(βq) = Eλ 12(βq)A λ,12 vc,βk(βq). (A14)

Comparing Eq. (A14) and Eq. (A13), we see that being

Ec(βk) = Ec(k) (the layer is invariant under rotation of±60◦),

the Hamiltonian in Eq. (A14) is the same as in Eq. (A13). This results in the following property for the energies Eλand coefficients Aλof the CT excitonic state:

E21λ(q)= E12λ(βq), (A15)

Aλ,vc,k21(q)= Aλ,vc,12βk(βq). (A16)

We now analyze the intralayer Frenkel exciton. In this case, the excitonic state is

| (q) = √1 N  λ,Ri ciλ(q) R,iλ (q), (A17) where  R,iλ (q)= k

Aλ,ivc,k(q)ack(i)Rib

vk(i)+q(i)Ri|0. (A18) The electron and the hole of the excitonic pair in this case both belong to the same layer i. The coefficients Aλ,ivc,k(q) satisfy the following excitonic eigenvalue equation:

 Eci(k(i))− Evi(k (i)+ q(i) )Aλ,ivc,k(q) + k  ¯

vRiRiRi,Ri(vk(i)+ q(i)ck(i)vk(i)+ q(i)ck(i))

− WRi,Ri

RiRi (vk(i)+ q(i)ck(i)vk

(i)

+ q(i)ck(i))Aλ,i vc,k(q)

= Eλ i(q)A

λ,i

Figure

FIG. 1. The real and imaginary parts of the macroscopic dielectric function Re M and Im M and the loss function −Im M− 1 calculated at two different wave vectors q along the in-plane M direction
FIG. 2. Exciton eigenvalue spectrum E λ (q) in bulk hBN for in-plane q along M for (a) the electron-hole correlation function ¯ L corresponding to Im M (q,ω) featuring singlet excitons, (b) the electron-hole correlation function L corresponding to the loss
FIG. 4. Dispersion of (a) singlet and (b) triplet excitons in the BN single layer. The color key is the same as in Fig
FIG. 6. Energy difference between the plasmon and the visible A + exciton as a function of momentum q for different interlayer distances.
+3

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The brackets indicate the fine structure splitting in the acceptor and bound exciton states (see the text for more details)... E loc , which was experimentally obtained from

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2 we present the spectra of bulk hexagonal BN, of the single sheet of hexagonal BN, and of different BN nanotubes with diameters ranging from 2.8 A ˚ (for the purely hypothetical

Taking into account few-layer systems, we investigate theoretically the effects of the number of layers on quasiparticle energies, absorption spectra, and excitonic states,

We present a detailed discussion of the electronic band structure and excitonic dispersion of hexagonal boron nitride (hBN) in the single layer configuration and in three

The binding energy of this peak increases strongly as the dimensionality is reduced from the 3-D bulk over the 2-D sheet to the 1-D tubes.. At the same time the quasi-particle band

Dynamic structure factor along three high symmetry lines in the first Brillouin zone ΓA, ΓK, and ΓM calculated in the GW-RPA (dashed lines) and from the solution of the BSE

By combining spatially resolved cathodoluminescence (SR-CL) spec- troscopy with a structural analysis of the crystal by means of Transmission Electron Microscopy (TEM) and Selected